We determine a set of generators for the Brunnian braids on a general surface M for M 6 D S 2 or RP 2 . For the case M D S 2 or RP 2 , a set of generators for the Brunnian braids on M is given by our generating set together with the homotopy groups of a 2-sphere.57M07, 57M99; 20F36, 55Q40

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It is shown for the first time that the photoassisted reduction of NO by CO into N2 and N2O can occur on TiO2 catalysts upon visible light irradiation (λ > 380 nm) at room temperature. The selectivity of photoreduction of NO into N2 reaches 90−95%. The CO2 formed which predominantly remains on the surface can be quantitatively desorbed after completion of the photoreaction by heating TiO2 to ∼500 K. The rates of NO consumption and product accumulation remain virtually constant upon successive admissions of the CO−NO mixture thus indicating a high stability of catalyst activity. It is found that the quantum yield of NO photoreduction by CO is considerably greater for visible light irradiation (λ = 405 + 436 nm) than for UV irradiation (λ = 365 nm). Experiments with 18O-enriched NO revealed that, under visible light irradiation, an intense oxygen isotopic exchange between NO and TiO2 develops. The photocatalytic reaction requires the presence in nonstoichiometric TiO2- x of electron-donor centers (Ti3+ ions, F and F+ centers) able to absorb visible light. A reaction mechanism is proposed which includes the following stages:  (a) NO photoadsorption along two parallel routes, by e- capture from the electron-donor center by NO to yield adsorbed NO- species and by NO interaction with the hole O- to give adsorbed nitrite NO2 -; (b) reduction of NO- into N2O and further into N2; and (c) photoreduction of adsorbed nitrite NO2 - into N2O and further into N2 by CO which regenerates the donor centers. It is assumed that the photoinduced oxygen heteroexchange proceeds via adsorbed nitrite complexes.

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We recall a group-theoretic description of the first non-vanishing homotopy group of a certain (n+1)-ad of spaces and show how it yields several formulae for homotopy and homology groups of specific spaces. In particular we obtain an alternative proof of J. Wu's group-theoretic description of the homotopy groups of a 2-sphere. The case n = 3Suppose that L, M, N are normal subgroups of a group G. This data gives rise to a commutative cube of spaces

The main purpose of this paper is to extend our knowledge of the derived functors of certain basic nonadditive functors. The discussion takes place over the integers, and includes a functorial description of the derived functors of certain Lie functors, as well as that of the main cubical functors. We also present a functorial approach to the study of the homotopy groups of spheres and of Moore spaces M.A; n/, based on the Curtis spectral sequence and the decomposition of Lie functors as iterates of simpler functors such as the symmetric or exterior algebra functors. As an illustration, we retrieve in a purely algebraic manner the 3-torsion components of the homotopy groups of the 2-sphere in low degrees, and give a unified presentation of the homotopy groups i .M.A; n// for small values of both i and n.18G55, 18G10; 54E30, 55Q40

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